1 Answer
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Step1. If $a_1,a_2,\alpha,\beta,\gamma$ are positive real numbers and $\gamma=\alpha+\beta$ holds,
$$\frac{\gamma^2}{a_1+a_2}\leq \frac{\alpha^2}{a_1}+\frac{\beta^2}{a_2}$$
holds too, since it is equivalent to $(\alpha a_2-\beta a_1)^2\geq 0$.

Step2. If $a_1,a_2,\alpha,\beta,\gamma,\delta$ are positive real numbers and $\delta=\alpha+\beta+\gamma$ holds,
$$\frac{\delta^2}{a_1+(a_2+a_3)}\leq \frac{\alpha^2}{a_1}+\frac{(\beta+\gamma)^2}{a_2+a_3}\leq\frac{\alpha^2}{a_1}+\frac{\beta^2}{a_2}+\frac{\gamma^2}{a_3}$$
holds too, in virtue of Step2. By induction, it is easy to prove the analogous statement for $k$ variables $a_1,\ldots,a_k$. In fact, this is useless to the proof, but quite interesting in itself :)

In fact, it is much easier to prove the stronger inequality: $$\sum_{k=1}^{n}\frac{2k+1}{a_1+\ldots+a_k}\leq -\frac{n^2}{a_1+\ldots+a_n}+\sum_{k=1}^{n}\frac{4}{a_k}.$$
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Jack D'AurizioOct 30 '12 at 9:17